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4PM1/01R Jun 2020, Question 11: Vector Geometry

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 13 marks

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The question, in short

In triangle OXY with OX = 2a and OY = 3b, A is midpoint of OX, B divides OY in ratio 1:2, and lines XB and AY meet at Z. Find vector expressions.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (a) · 1 mark

Express vector AB in terms of a and b, simplified.

  1. 1OA = a, OB = b
  2. 2AB = OB − OA = b − a

AnswerAB = b − a

Part (b) · 9 marks

Use a vector method to determine OZ as a simplified expression in a and b.

  1. 1OZ = OB + λBX = b + λ(2a − b) = (1 − λ)b + 2λa
  2. 2OZ = OA + μAY = a + μ(3b − a) = (1 − μ)a + 3μb
  3. 3Compare: 2λ = 1 − μ and 1 − λ = 3μ
  4. 43(1 − 2λ) = 1 − λ ⇒ λ = 2/5 (μ = 1/5)
  5. 5OZ = (4/5)a + (3/5)b = (1/5)(4a + 3b)

AnswerOZ = (1/5)(4a + 3b)

Part (c) · 3 marks

Given M lies on XY with O, Z, M collinear, find OM in terms of a and b.

  1. 1OM = p × (1/5)(4a + 3b) and OM = 2a + q(−2a + 3b)
  2. 24p/5 = 2 − 2q and 3p/5 = 3q
  3. 3p = 5/3
  4. 4OM = (1/3)(4a + 3b)

AnswerOM = (1/3)(4a + 3b)

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